Phillip Sprangle Naval Research Laboratory Washington, DC Northeastern University Boston February 7, 2003 Collaborators

1. and “Stopped Light” “Faster than Light” Propagation of Laser Pulses Phillip Sprangle Naval Research Laboratory Washington, DC Northeastern UniversityBostonFebruary 7, 2003 Collaborators : J. R. Peñano and B. Hafizi

2. Outline • Introduction • Phaseand Group Velocities • “Faster than Light” Experiment • Interpretation of “Faster than Light” Experiments • “Stopped” Light • Summary

3. ”Faster than Light” Propagation of Laser Pulses • In a recent Nature article [1], researchers reported observing • superluminalpropagation of a laser pulse in a gain medium • by a new mechanism • The Nature article received a great deal of attention in both • the scientific community and the world press [1] L. J. Wang, A. Kuzmich, and A. Doggariu, Nature (London)406, 277 (2000)

5. World Wide Popular Press Releases • “The speed of light is exceeded in lab”, Washington Post, 20 July, 2000 • “Speed of light may not be the last word”, The Boston Globe, 20 July, 2000 • “Faster than light, maybe, but not back to the future”, New York Times, 30 July, 2000 • “A pulse of light breaks the ultimate speed limit”, Los Angeles Times, 20 July, 2000 • “Light goes backwards in time”, The Guardian, 20 July, 2000 • “It’s confirmed: speed of light can be broken”, India Today, 21 July, 2000 • “Ray of light for time travel”, South China Morning Post, 21 July, 2000

6. signal velocity u Causality • Causality requires that the effect appears after the cause • According to special relativity, in a frame moving • with velocity v < c, the time interval between events is • Special relativity requires that the signal velocity be less than c • Causality is violated if signal velocity u is greater than c

7. Cesium gas T = 3.7sec ( 3700 ft) = 6cm laser frequency is between two gain lines “Faster than Light” Experiment Wang, et al., Nature, 406, 277 (2000) Time delay ׃

8. Experimentally observed advancement of a laser pulse in a gain doublet[Wang, et al., Nature, 406, 277 (2000)] The Nature article claims that 1) “differs from previously studied anomalous dispersion associated with an absorption or a gain resonance” 2) “ the shape of the pulse is preserved” 3) “the argument that the probe pulse is advanced by amplification of its front edge does not apply”

9. Laser Pulse vg vph Phase and Group Velocity of Laser Pulses Phase Velocity is the velocity of the phase Group Velocity is the velocity of the pulse envelope • If the pulse envelope does not distort then the group velocity is the • velocity of energy flow

10. Two Level Atom, α > 0 Refractive Index • The refractive index represents the medium’s response to the fields • Using the Classical Lorentz or Quantum Two Level Atom model where α denotes population inversion, N is the density of atoms, Ω is the atomic binding frequency, Γ is the damping rate , q is the electronic charge and m the electronic mass

11. Phase velocity: where • Group velocity: Phase and Group Velocities • There are conflicting interpretations of how pulses propagate • when vg is “abnormal” , i.e., vg < 0 or vg > c

12. Phase and Group Velocities

13. Abnormal Group Velocities Can have abnormal group velocities, vg > c or vg < 0, in a gain medium and/or loss medium

14. Two Level Atom , α < 0 • Index of refraction: • For low damping (amplifying) Superluminal Group Velocity (“Optical Tachyons”)

15. Absorptive Medium Excited Medium Im(Dn)103 Im(Dn)103 Abnormal Group Velocities

16. Laser electric field ׃ , Interpretation of “Faster than Light” Experiments Sprangle, Peñano and Hafizi, Phys. Rev. E, 64, 026504 (2001) Pulse Envelope equation ׃

17. distorted envelope differential gain higher order differential gain initial envelope vg = c/(1 + c1) Solution of Envelope Equation Laser pulse envelope: • Front pulse propagates at the speed of light and undergoes distortion • Front is preferentially amplified when k1 < 0, i.e., differential gain.

18. Pulse Distortion in a Gain Medium differential gain Nature Article |A(z = L,t)| vacuum (v = c) t - z/c Front of Pulse

19. Group Velocity Laser Spectrum vg(w)/c Laser Envelope Gain Spectrum Laser Spectrum A(z,t)/A0 -Im(Dn) Apparent Superluminal Propagation of a Laser Pulse W= 0.08 ωo , G = 0.08 ωo

20. Differential Gain Interpretation Wang, et al. DT = 62 nsec pulse peak pulse amplitude Time (msec) Comparison with Experiment Using the reported experimental parameters, the differential gain effect can account for the observed 62 nsec pulse advancement Pulse advancement occurs by amplification of the leading edge

21. Superluminal Laser Pulses • Wang et al. interpreted experimental results as superluminal • propagation without pulse distortion • Analysis shows that pulse distortion was present and • responsible for the apparent superluminal propagation • The front of the distorted pulse propagates at speed of light • while the peak was superluminal

22. “Stopped Light” Pulses • Quantum interference effects can lead to • Electromagnetically Induced Transparency* (EIT) • EIT results in • Extremely low group velocities • High transparency • Refractive index is unity * S. E. Harris , Phys. Rev. Lett. 72, 52 (1994)

23. 3 Strong coupling beam Highly damped level 2 Probe pulse 1 >> 1 = 1 Electromagnetically Induced Transparency 3 Level Atom • Quantum interference effects result in state 3 being unpopulated, • no interaction with probe beam

24. Probe pulse Ultra cold gas of Sodium atoms Bose-Einstein Condensate Strong coupling beam Giant Kerr nonlinearity ׃ “Stopped Light” Experiment S.E. Harris Phys. Today 50, 36 (1997), L. V. Hau, et.al., Nature 397, 594 (1999) Probe pulse velocity reduced to 38 mph

25. Probe pulse ׃ • Envelope of the probe pulse at resonance : Rabi frequency of coupling beam Probe Pulse has Extremely Low Group Velocity and Losses • The group velocity of the probe pulse is • No losses in probe pulse

26. Highly Opaque Material (no coupling beam) Wc/Wp = 0, Dw /Wp= 1, G3/Wp = 10

27. Ultra Slow Light (with coupling beam) Probe pulse compressed by the factor c/vg >> 1 The excited states are not populated The probe pulse energy is converted into the coupling beam energy Wc /Wp = 1/40, Dw = 0

28. Quantum interference effects (EIT) can result in • - extremely low group velocities, vg ~ 38 mph • extremely high transparencies Summary • Recent experiments have been interpreted as superluminal laser pulse • propagation [Wang, et al., Nature, 406, 277 (2000)] • The apparent superluminal propagation is actually due to a • differential gain mechanism [Sprangle, et al., Phys. Rev. E, 64,026504 (2001)] • which can give the appearance of faster-than-light propagation • The leading edge of laser pulse travels at speed of light in vacuum • Differential pulse distortion can occur in a gain or a loss medium • Potential applications include : optical switching and quantum computing

29. Back–Up View Graphs

30. Laser Envelope Equation Wave Equation bound electron Representation of electric and polarization fields, 1-D Need relationship between A(z,t) and B(z,t) Causal relationship between P and E through susceptibility

31. Electromagnetic Energy flow z laser pulse envelope where ׃carrier frequency, ko׃ carrier wavenumber and unit vector Electromagnetic Wave Equation Wave Equation: where P is the polarization field representing the medium response Electric Field:

32. Group Velocity vg dispersive medium z : refractive index Expand wavenumber about carrier frequency where is the group velocity

33. Assuming low damping • The real part of the refractive index near the • resonance frequency, • If the medium is amplifying ( < 0) then the • group velocity is superluminal ( vg > c), (“Optical Tachyons”) Superluminal Group Velocity

34. Laser Envelope Equation In frequency domain Using the representations for the fields Since is non zero for small positive values of

35. Laser Envelope Equation By definition and therefore The refractive index is given by

36. Standard Fourier Transform Approach In the standard Fourier Transform approach care must be taken in ordering the terms For example: Index is expanded to 1st order (GVD is neglected) Exponential factor contains contributions beyond the order of approximation

37. Lowest Order Fourier Transform Solution The lowest order solution ( neglecting k2 ) is Pulse propagates undistorted with group velocity: If vg > c, or < 0 ( abnormal ) the result is unphysical

38. Gaussian Shaped Laser Pulses Using the standard Fourier Transform approach, GVD effects , can be analyzed for specific pulse shapes (e.g., Gaussian) For an initially Gaussian pulse: where T is the initial pulse duration Pulse envelope propagates at velocity vg and remains Gaussian, but with a different amplitude and width. [C.G.B. Garrett & D.E. McCumber, PRA, 1, 305 (1970), E.L. Bolda, et al., PRA, 49, 2938 (1994)]

39. Gaussian Laser Pulse Solution Input pulse is Gaussian: The pulse at z is distorted (not Gaussian):

40. Comparison with Experiments Gain Medium Susceptibility model describing a gain doublet [Wang, et al., Nature, 406, 277 (2000)] : For small line width, i.e., , A gain medium (inverted population) implies that k1 < 0 , ( vg < 0 or > c )

41. simulation 2nd order solution vacuum solution Exact Numerical Solution front Input pulse shape: |A(z = L, t)| peak |A(z = L, t)| vacuum |A(z = L, t)| back |A(z = L, t)| t / 2T t / 2T

42. Comparison with Experiment Parameters Numerical calculation • M1,2 = M = 0.18 Hz • f1 = 3.5 x 1014 Hz • f2 = f1 + 2.7 MHz • = 0.46 MHz L = 6cm Wang, et al. vp(w0) = -c/305

43. Pulse Energy For the example:

44. Nonlinear Lorentz Model of Excited Atoms Displacement of electronic distribution (Duffing Eq.) Polarization field:

45. Index of refraction inverted population term for no homogeneous gain, but can have differential gain

46. Background In a gain medium, superluminal propagation proceeds by the preferential amplification of the leading edge of the pulse. C.G.B. Garrett & D.E. McCumber, PRA, 1, 305 (1970) Propagation of Gaussian pulses. R.Y. Chiao, PRA, 48, R34 (1993) Off-resonance superluminal propagation A.M. Steinberg & R.Y. Chiao, PRA, 49, 2071 (1994) Dispersionless propagation in a gain doublet E.L. Bolda, et al., PRA, 49, 2938 (1994) (what’s new about this?)

47. Background In an attenuating medium, undistorted, superluminal, and causal propagation of pulses (not necessarily Gaussian) is possible under certain conditions. [M.D. Crisp, PRA 4, 2104 (1971)]

48. Envelope Equation Substituting E and P, and relationship between A(z,t) and B(z,t), into wave equation yields an envelope equation For narrow spectral pulse widths ( long pulses ) can keep terms to order ( lowest order in GVD ) Envelope Equation

49. Analysis Consistent Solution: Exponential is expanded to consistent order Pulse propagates at the speed of light and undergoes distortion. Front is preferentially amplified when k1 < 0, i.e., differential gain.

50. Analysis Integral solution: Standard Approach: Lowest order solution is obtained by neglecting k2 and carrying out the integral without expanding the exponential: [Refs.] Analysis results in a pulse propagating undistorted at the group velocity: